A nonsmooth Lyapunov function and stability for ODE’s of Caratheodory type

Almost Everywhere Conditions for Hybrid Lipschitz Lyapunov Functions

“…De to to conditions 3) and 5) of Definition 3, for each y ∈ S there exists a unique i ∈ I such that V (y) = V i (y) and ∇V (y) = ∇V i (y). The conclusion follows from (9).…”

“…which implies (9). On the other hand, one can see that every flowing solution starting at some x 0 ∈ R := {x ∈ R 2 | x 1 = x 2 }, x 0 = 0 goes to infinity sliding along R. See Figure 1 for a graphical representation.…”

“…Corresponding author: mdellaro@laas.fr Work supported in part by ANR grants CONVAN (number ANR-17-CE40-0019-01) and HANDY (number ANR-18-CE40-0010). approach is presented in [9], where the nonsmooth function is studied exploiting regularization-via-convolution technique.…”

Construction of nonpathological Lyapunov functions for discontinuous systems with Caratheodory solutions

“…A nonpathological Lyapunov theorem to construct nonpathological Lyapunov function (NPLF) for discontinuous time invariant dynamical systems with Caratheodory solutions is proposed in 1. The notion of Caratheodory solutions is accepted as a good notion of solution for discontinuous systems 1–3. In this paper, we take advantage of a notion of derivative for locally Lipschitz continuous functions which are nonpathological introduced in 4, and then improved and applied by 1.…”

A Converse Liapunov Theorem for Uniformly Locally Exponentially Stable Systems Admitting Carathéodory Solutions

“…A recent article by Grzanek et al (2008) uses a general form of the integral theorem of calculus to guarantee (Liapunov) stability of solutions, often the most difficult step in all the above results, and go on to prove asymptotic stability for general Carathéodory solutions of system with a Liapunov function. Aeyels and Peuteman (1998) have studied stability of systems using an integral formulation of the Liapunov decent condition.…”